[seqfan] The anti-Carmichael numbers

[Tomasz Ordowski]

Dear SeqFans!

Let's define the "anti-Carmichael numbers":

Numbers k > 1 such that p-1 does not divide k-1 for every prime p dividing
k.

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247,
253, 275, ...

The number k > 1 is an anti-Carmichael if and only if gcd(k,b^k-b) = 1 for
some integer b.

Are there anti-Carmichael numbers k such that gcd(k,b^k-b) > 1 for all
natural b < lpf(k)?

Best regards,

Thomas
__________
The number k > 1 is an anti-Carmichael pseudoprimes to base a
if and only if a^k == a (mod k) and gcd(k,b^k-b) = 1 for some anti-base b.

Cf. http://oeis.org/history/view?seq=A316907&v=38
and http://oeis.org/history/view?seq=A300762&v=14